に (For)

This section needs some clean up and encompasses a few different things, so I’ve included a summary block.

Summary:
に can mean for
に can be used for potential verbs
には is は marking に as part of the topic.
には tends to have more emphasis on the meaning of に (location, direction, etc)

に, when attached to a noun may mean “for”. This に marks a person/thing as a “place” for the verb. 
With people, it could be thought to refer to that person’s abilities, as in “Within that person’s abilities”. 

It is typically used for marking the topic of a need, or for marking the topic of a potential (see potential に). You may interpret this is a “direction” of a need or potential. 
For simplicity, you can think of it as “For“. 

• 私に何か用はある？
  Do you need anything from me. 
  
  
• 勉強に必要なもの
  Things needed for studying. (Things are needed in studying).(see also 必要)
  
  

Potential に

With potential form verbs, に can be used with or in place of は.  You’ll hear this a lot. It still means something like “for” or “by” here.

• 僕にわかる
  For me, understandable. -> Understandable for me.
  
  
• 彼にできる
  For him, doable. -> Doable for me.
  
  
• 大人に飲める
  For adults, drinkable. 
  
  
• 彼女にならできる
  If its for her, she can do it. 
  
  

This に, while specific to a particular person, doesn’t exclude others from being able to do it. 

This is (apparently) “location marker” に、marking an “location” where it is possible.
Source 1, Source 2, 

If you want to learn more about this topic, see the nuance section at the bottom. 

には

に as the Topic

“For” Emphasis

には makes に part of the topic. It puts emphasis on the に part. 

It’s typically used to stress that something is possible for a specific person or thing (Potential に).
Third person applications of the potential に above are usually become には。
Other meanings of に may also be used (Location, etc.)

• 私にはわかる
  I understand. (I understand, maybe not someone else) (Potential に)
  
  
• 彼にはできる
  He can do it. (Potential に)
  
  
• 城にはお姫様が住んでいる
  There’s a princess living in the castle. (Castle is topic and location). (Location に)
  
  
• 公園にはブランコやジャングルがあるんだ
  At the park, there’s swings and jungle gyms. (location に)
  
  

For purposes of

For purposes of
Dictionary form verb + には

には may also be used with a dictionary form verb to mean “in order to” or “for the purpose of”.

• 飲むには最適だ
  The best for drinking. 
  
  
• 泳ぐにはちょうどいい温度
  The perfect temperature for swimming. 
  
  
• 勉強するには最高だ
  It’s the best for studying. 
  
  
• 男には大事なこと
  Its important for men. 
  
  

に and が With Potential Verbs

This section is on nuance. If you don’t care, skip it. It’s probably better to figure it out by experience anyway. 
What’s the difference between に and が? There’s a lot of research on this, but here’s the gist:

• Use に most of the time. 
• You almost definitely will use に if が is used with the potential verb. (see below). 
• Use が if the statement focuses on a specific person.
• Use が if you are responding to a question or comparing people. 
• が should be used with the thing that is doable most of the time
• に should be used with the thing capable of doing most of the time
• に places emphasis on the potential phrase
• が places emphasis on the actor

Consider:

• ご飯が食べれる – (By Someone) Rice can be eaten. 
• 私が食べれる – I (it) can be eaten. – Ok as a response to a question. 
• 私に食べれる – It’s eatable by me. 
• 私がご飯が食べれる – I can food can be eaten. – (Doesn’t this just sound bad?)

And finally consider:

• 私にできることは – Things that are doable by me (This sounds like “only these things”)
• 私ができることは – I can do…(This sounds like a non exhaustive list, general statement).

Source 1
Source 2
Unused additional Japanese Reading